Timothy Robin Teng†

November 23, 2011 Abstract In this paper we review the hedging of interest rate derivatives priced under a risk-neutral measure, and we compute self-financing hedging strategies for various derivatives using the Clark-Ocone formula.

Key words: Bond markets, hedging, infinite-dimensional analysis, Clark-Ocone formula, swaptions, bond options, caplets. Mathematics Subject Classification: 91B28, 60H07.

1

Introduction

While the pricing of interest rate derivatives is well understood, due notably to the use of the change of numeraire technique, the computation of hedging strategies for such derivatives presents several difficulties. In general, hedging strategies appear not to be unique and one is faced with the problem of choosing an appropriate tenor structure of bond maturities in order to correctly hedge maturity-related risks, see e.g. [3] in the jump case. In [6], self-financing hedging strategies have been computed for swaptions in a geometric Brownian model, using the associated forward measure. In [7] this approach has been extended to other interest rate derivatives using the Markov property and stochastic integral representation formulas under change of numeraire, which is a natural tool for the pricing of such derivatives. ∗

Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, SPMS-MAS, 21 Nanyang Link, Singapore 637371. [email protected] † Department of Mathematics, Ateneo de Manila University, Loyola Heights, Quezon City, Philippines. [email protected]

1

In this paper we focus on the hedging of interest rate derivatives under the risk-neutral probability measure P itself, using the general framework for the hedging of interest rate derivatives introduced in [1], [2], which is based on a cylindrical Wiener process (Wt )t∈IR+ with values in a Hilbert space H under P. In particular, we compute hedging strategies for interest rate derivatives, using both Delta hedging and the Clark-Ocone formula. As in [7], we determine the relevant tenor structure from payoff structure of the claim, in such a way that the hedging strategy does not explicitly depend on bond volatilities. We proceed as follows. The notation on bond markets and option pricing under the risk-neutral measure is introduced at the end of this Section. In Section 2 we derive self-financing hedging strategies for interest rate derivatives based on the Markov property. Finally in Section 3 we use the Clark-Ocone formula to compute selffinancing hedging strategies for interest rate derivatives. We mainly consider three examples, namely swaptions, bond options, and caplets on the forward and LIBOR rates.

Notation We work in the infinite dimensional framework of [1], [2]. Consider a probability space (Ω, F, P) on which is defined a cylindrical Wiener process (Wt )t∈IR+ with values in a Hilbert space H. The measure P is taken as a risk-neutral measure. Let (rt )t∈IR+ denote a short term interest rate process adapted to the filtration (Ft )t∈IR+ generated by (Wt )t∈IR+ , consider the bank account process Bt = e

Rt 0

rs ds

,

t ∈ IR+ .

By risk-neutral valuation under the measure P, an FT -measurable claim with payoff ξ, maturity S and exercise date T , is priced at time t as i h RS − t rs ds ˜ IE e BS ξ Ft = Bt IE[ξ˜ | Ft ], 0 ≤ t ≤ T < S, where ξ˜ = BS−1 ξ ∈ L1 (P, FS ) 2

(1.1)

denotes the discounted payoff of the claim. We will work with a continuous Ft -adapted asset price process (Xt )t∈IR+ taking values in a real separable Hilbert space F of real-valued functions on IR+ , usually a weighted Sobolev space F of real-valued functions on IR+ , cf. [5] and § 6.5.2 of [2]. In the sequel, (Xt )t∈IR+ will represent either a bond price curve taking values in the function space F , or a real-valued asset price when F = IR, cf. also [7]. We note that the discounted asset price ˜ t := Xt , X 0 ≤ t ≤ T, Bt is an F -valued martingale under the risk-neutral measure P, provided it is integrable ˜ t )t∈IR+ satisfies under P. More precisely we will assume that (X ˜ t = σt dWt , dX

t ∈ IR+ ,

(1.2)

where (σt )t∈IR+ is an LHS (H, F )-valued adapted process of Hilbert-Schmidt operators from H to F .

2

Risk-neutral hedging in bond markets

Assume that the discounted claim ξ˜ ∈ L2 (Ω) has the predictable representation Z T ˜+ ˜ t iF ∗ ,F , ξ˜ = IE[ξ] hφ˜t , dX (2.1) 0

˜ t )t∈[0,T ] is given by (1.2) and (φ˜t )t∈[0,T ] is a square-integrable F ∗ -valued where (X adapted process of continuous linear mappings on F under P, from which it follows that the discounted claim price V˜t := IE[ξ˜ | Ft ],

0 ≤ t ≤ T,

is a martingale that can be decomposed as Z t ˜ ˜ ˜ s iF ∗ ,F , Vt = IE[ξ] + hφ˜s , dX

0 ≤ t ≤ T.

(2.2)

0

Consider a portfolio strategy (φ˜t , η˜t ) with value Z ∞ ˜ Vt := hφt , Xt iF ∗ ,F + η˜t Bt = Xt (y)φ˜t (dy) + η˜t Bt T

3

(2.3)

where φ˜t (dy) will denote the amount of bonds having maturity in [y, y + dy] in the portfolio, and η˜t denotes the quantity invested in the money market account in the portfolio at time t ∈ [0, T ]. Definition 2.1 The portfolio strategy (φ˜t , η˜t ), 0 ≤ t ≤ T , is said to be self-financing if dVt = hφ˜t , dXt iF ∗ ,F + η˜t dBt ,

0 ≤ t ≤ T.

(2.4)

We say that the portfolio strategy (φ˜t , η˜t ) hedges the option with payoff ξ if for all t ∈ [0, T ], its value hφ˜t , Xt iF ∗ ,F + η˜t Bt satisfies i h RS − t rs ds ˜ ∗ ξ Ft , hφt , Xt iF ,F + η˜t Bt = IE e i.e. ˜ t iF ∗ ,F + η˜t = V˜t = hφ˜t , X

Z

∞

˜ t (y)φ˜t (dy) + η˜t , X

0 ≤ t ≤ T,

T

˜ t = Bt−1 Xt , 0 ≤ t ≤ T , is the discounted asset price. where X The next proposition is well known and is a particular case of a general change of numeraire argument, cf. e.g. [6], [7]. ˜ t , φ˜t iF ∗ ,F , 0 ≤ t ≤ T , where (φ˜t )t∈[0,T ] satisfies Proposition 2.2 Letting η˜t = V˜t − hX (2.1), the portfolio (φ˜t , η˜t )t∈[0,T ] with value Vt = hφ˜t , Xt iF ∗ ,F + η˜t Bt ,

0 ≤ t ≤ T,

(2.5)

˜ is self-financing and hedges the claim ξ = BS ξ. Proof. By (2.5) we have i h RS − t rs ds ˜ ˜ Vt = Bt Vt = Bt IE[ξ | Ft ] = IE e ξ F t ,

0 ≤ t ≤ T,

˜ Next, we show that it is hence the portfolio (φ˜t , η˜t )t∈[0,T ] hedges the payoff ξ = BS ξ. self-financing. We have dVt = d(Bt V˜t ) = V˜t dBt + Bt dV˜t + dBt · dV˜t 4

˜ t iF ∗ ,F + dBt · hφ˜t , dX ˜ t iF ∗ ,F = V˜t dBt + Bt hφ˜t , dX ˜ t iF ∗ ,F dBt + Bt hφ˜t , dX ˜ t iF ∗ ,F + dBt · hφ˜t , dX ˜ t iF ∗ ,F + (V˜t − hφ˜t , X ˜ t iF ∗ ,F )dBt = hφ˜t , X ˜ t )iF ∗,F + (V˜t − hφ˜t , X ˜ t iF ∗ ,F )dBt = hφ˜t , d(Bt X = hφ˜t , dXt iF ∗ ,F + η˜t dBt . where ˜ t iF ∗ ,F = 0. dBt · dV˜t = dBt · hφ˜t , dX Next we recall how the process (φ˜t )t∈IR+ in the predictable representation (2.2) can be computed by the Clark-Ocone formula, cf. [1]. Let D denote the Malliavin gradient defined on smooth functionals of Brownian motion of the form ξ˜ = f (Wt1 (h1 ), . . . , Wtn (hn )),

0 < t1 < · · · < tn ,

f ∈ Cb1 (IRn ), h1 , . . . , hn ∈ H, as Dt ξ˜ =

n X k=1

1[0,tk ] (t)

∂f (Wt1 (h1 ), . . . , Wtn (hn )) ⊗ hk , ∂xk

(2.6)

cf. § 5.1.2 of [2], and extended by closability to its domain Dom (D). To hedge a claim ξ in this setting, we decompose the discounted payoff ξ˜ as Z T Z T ˜ ˜ ˜ ˜ ξ = IE[ξ] + hIE[Dt ξ | Ft ], dWt iH = IE[ξ] + hαt , dWt iH 0

(2.7)

0

where, by the Clark-Ocone formula, αt = IE[Dt ξ˜ | Ft ],

(2.8)

cf. Theorem 5.3 of [2]. From Relations (1.2) and (2.7) the process (φ˜t )t∈IR+ in (2.1) is given by φ˜t = (σt∗ )−1 αt = (σt∗ )−1 IE[Dt ξ˜ | Ft ], provided σt∗ : H → F is invertible, 0 ≤ t ≤ T . 5

0 ≤ t ≤ T,

(2.9)

Markovian case ˜ t )t∈IR+ has the Markov property, and the dynamics Next, assume in addition that (X ˜ t = σt (X ˜ t )dWt , dX

(2.10)

where x 7→ σt (x) ∈ LHS (H, F ) is a Lipschitz function from F into the space of Hilbert-Schmidt operator from H to F , uniformly in t ∈ IR+ . In case H = F = IR ˜ t ) = σ(t)X ˜ t , i.e. the martingale (X ˜ t )t∈[0,T ] is a geometric Brownian motion and σt (X under P with deterministic variance (σ(t))t∈[0,T ] . ˜ T can be computed as In the Markovian setting of (2.10), Dt X ˜ T = Y˜t,T σt (X ˜ t ), Dt X

(2.11)

where (Y˜t,T )t∈[0,T ] is solution of Z t ˜ u )Y˜s,u dWu , Y˜s,t = Id + ∇σu (X

0 ≤ s ≤ t,

(2.12)

s

˜ T ) we get, assuming that g˜ is Lipschitz, cf. Proposition 6.7 of [2], hence in case ξ˜ = g˜(X ˜ t ))−1 αt φ˜t = (σt∗ (X i h ˜ t ))−1 IE Dt g˜(X ˜ T ) Ft = (σt∗ (X i h ∗ ˜ −1 ∗ ˜ ˜ = (σt (Xt )) IE (Dt XT ) ∇˜ g ( XT ) F t i h ˜ t ))−1 IE (Y˜t,T σt (X ˜ t ))∗ ∇˜ ˜ T ) Ft = (σt∗ (X g (X i h ∗ ˜ T ) Ft , = IE Y˜t,T ∇˜ g (X 0 ≤ t ≤ T, cf. [2] § 6.5.5. The use of (2.8) can be somewhat limited since the application of D to ξ˜ can lead to technical difficulties due to the differentiation of B −1 = P˜S (S), and (2.12) can be S

difficult to solve. In the remaining of this section we take T = S. 6

European options We close this section with an application of the Delta hedging method to European ˜ T ) where g˜ : F → IR and (X ˜ t )t∈IR+ has type options with discounted payoff ξ˜ = g˜(X the Markov property as in (2.10), where σ : IR+ × F → LHS (H, F ). ˜ T ) is priced at time t as In this case the option with payoff ξ = BT g˜(X i i h RT h ˜ T ) Ft = Bt C(t, ˜ X ˜ t ), IE e− t rs ds ξ Ft = Bt IE g˜(X ˜ x) on IR+ × F . However this formula allows one to for some measurable function C(t, deal with only a limited range of options, such as exchange options. ˜ x) is C 2 on IR+ × F , we have the following corollary Assuming that the function C(t, of Proposition 2.2. ˜ X ˜ t ) − h∇C(t, ˜ X ˜ t ), X ˜ t iF ∗ ,F , 0 ≤ t ≤ T , the portfolio Corollary 2.3 Letting η˜t = C(t, ˜ X ˜ t ), η˜t )t∈[0,T ] with value (∇C(t, ˜ X ˜ t ), X ˜ t iF ∗ ,F , Vt = η˜t Bt + h∇C(t,

t ∈ IR+ ,

˜ T ). is self-financing and hedges the claim ξ = BT g˜(X Proof.

This result follows directly from Proposition 2.2 by noting that by Itˆo’s

formula, cf. Theorem 4.17 of [4], we have ˜ X ˜ t ) = h∇C(t, ˜ X ˜ t ), dX ˜ t iF ∗ ,F dC(t, By the martingale property of V˜t under P and the predictable representation (2.2) we have ˜ t iF ∗ ,F dV˜t = hφ˜t , dX which ultimately gives us ˜ X ˜ t ), φ˜t = ∇C(t,

0 ≤ t ≤ T.

7

˜ T ) = (XT − κBT )+ As a consequence the exchange call option with payoff ξ = BT g˜(X ˜ t )t∈[0,T ] a geometric Brownian motion under P with (σ(t))t∈[0,T ] a deterministic on (X function, the option price is given by the Margrabe formula i h RT ˜ X ˜ t ) = Xt Φ+ (t, κ, X ˜ t ) − κBt Φ− (t, κ, X ˜ t ), IE e− t rs ds (XT − κBT )+ Ft = Bt C(t, (2.13) where the functions Φ+ (t, κ, x) and Φ− (t, κ, x) are defined as log(x/κ) v(t, T ) log(x/κ) v(t, T ) Φ+ (t, κ, x) := Φ + and Φ− (t, κ, x) := Φ − , v(t, T ) 2 v(t, T ) 2 where Z v(t, T ) =

T

|σ(s)|2 ds,

0 ≤ t ≤ T.

t

∂ C˜ By Corollary 2.3 and the relation (t, x) = Φ+ (t, κ, x), x ∈ IR, applied to the ∂x ˜ x) = xΦ+ (t, κ, x) − κΦ− (t, κ, x), the portfolio function C(t, ˜ t ), −κΦ− (t, κ, X ˜ t )), (φ˜t , η˜t ) = (Φ+ (t, κ, X

0 ≤ t ≤ T,

(2.14)

is self-financing and hedges the claim (XT − κBT )+ . ˜ T ) are not frequent and in In general, however, claim payoffs of the form BT g˜(X Section 3 we will use another method, i.e. the Clark-Ocone formula, to hedge interest rate derivatives. Note that the Delta hedging method requires the computation of the ˜ x) and that of the associated finite differences, and may not apply to function C(t, path-dependent claims.

3

Hedging by the Clark-Ocone formula

In this section we compute hedging strategies for interest rate derivatives via the Clark-Ocone formula, and we refer to [8] for the pricing computations not included here. We consider a real-valued Wiener process (Wt )t∈R+ under a risk-neutral probability measure P and we take (Xt )t∈IR+ = (Pt )t∈IR+ , i.e. the bond price curve (Pt )t∈IR+ takes values in a Sobolev space F of real-valued functions on IR+ , cf. [5] and [1] for 8

examples. Let µ ∈ F ∗ denote a finite measure on IR+ with support in [T, ∞), and consider the asset price Z

∞

Pt (µ) = hµ, Pt iF ∗ ,F =

Pt (y)µ(dy). T

In practice, µ(dy) and φ˜t (dy) will be finite point measures, i.e. sums of the form φ˜t (dy) =

j X

αk δTk (dy)

k=i

of Dirac measures at the maturities Ti , . . . , Tj of a given a tenor structure, in which αk (t) represents the amount allocated to a bond with maturity Tk , i ≤ k ≤ j, in which case (2.5) reads Vt =

j X

αk (t)Pt (Tk ) + η˜t Bt ,

0 ≤ t ≤ T,

k=i

We will assume that the dynamics of (Pt )t∈R+ is given by dPt = rt Pt dt + Pt ζt dWt ,

(3.1)

where (ζt )t∈[0,T ] is an LHS (H, F )-valued deterministic mapping, with dP˜t (y) = P˜t (y)ζt (y)dWt ,

(3.2)

i.e. we take σt (P˜t ) = ζt (·)P˜t (·) in (2.10). Consider a discounted payoff function of the form ξ˜ = g˜(P˜T (µ)), with maturity T , where g˜ : IR → IR is a Lipschitz function and Z ∞ PT (µ) = PT (x)µ(dx). T

The next result is stated for discounted payoffs. Proposition 3.1 Letting R T P (x) T − r ds 0 φ˜t (dx) = IE e t s g˜ (P˜T (µ)) Ft µ(dx), Pt (x) 9

(3.3)

˜ t , φ˜t iF ∗ ,F , 0 ≤ t ≤ T , yields a self-financing hedging portfolio hedging and η˜t = V˜t − hX the claim with payoff ξ = BT g˜(P˜T (µ)). Proof. We note that since ξ˜ = g˜(P˜T (µ)), we have Dt ξ˜ = Dt g˜(P˜T (µ)) = g˜0 (P˜T (µ))Dt P˜T (µ), where Dt P˜T (µ) =

Z

∞

ζt (x)P˜T (x)µ(dx).

T

Therefore the process (αt )t∈[0,T ] in (2.7) is given by h i αt = IE Dt ξ˜ | Ft h i 0 ˜ ˜ = IE g˜ (PT (µ))Dt PT (µ) | Ft Z ∞ h i 0 ˜ ˜ ζt (x) IE g˜ (PT (µ))PT (x) | Ft µ(dx), = T

hence Z

hαt , dWt iH

∞

h

i ˜ ˜ = IE g˜ (PT (µ))PT (x) | Ft µ(dx)ζt (x)dWt T # Z ∞ " P˜T (x) 0 ˜ IE g˜ (PT (µ)) = | Ft µ(dx)dP˜t (x). P˜t (x) T 0

From (2.7) the process (φ˜t )t∈[0,T ] in (2.1) is given by " # ˜T (x) P φ˜t (dx) = IE g˜0 (P˜T (µ)) Ft µ(dx), P˜t (x) and it remains to apply Proposition 2.2 with (Xt )t∈IR+ = (Pt )t∈IR+ .

Next, we apply Proposition 3.1 to swaptions. Swaptions on the LIBOR rate Consider a tenor structure {T ≤ Ti , . . . , Tj } and the swaption on the LIBOR rate with payoff ξ = (PT (Ti ) − PT (Tj ) − κP (T, Ti , Tj ))+ 10

(3.4)

where P (T, Ti , Tj ) =

j−1 X

τk PT (Tk+1 )

k=i

is the annuity numeraire, with τk = Tk+1 − Tk , k = i, . . . , j − 1. The next corollary follows from Proposition 3.1. Corollary 3.2 Letting "

# ˜ PT (Ti ) φ˜t (dx) = IE 1{S(T,Ti ,Tj )>κ} Ft δTi (dx) P˜t (Ti ) " # P˜T (Tj ) −(1 + κτj−1 ) IE 1{S(T,Ti ,Tj )>κ} Ft δTj (dx) P˜t (Tj ) # " j−1 X P˜T (Tk ) τk−1 IE 1{S(T,Ti ,Tj )>κ} −κ Ft δTk (dx), P˜t (Tk ) k=i+1 and η˜t = 0, 0 ≤ t ≤ T , yields a self-financing hedging portfolio hedging the claim with payoff (3.4), without any investment in the money market account, where S(T, Ti , Tj ) =

PT (Ti ) − PT (Tj ) P (T, Ti , Tj )

is the swap rate. Proof. We apply Proposition 3.1 with µ(dx) = δTi (dx) − δTj (dx) − κ

j−1 X

τk δTk (dx),

k=i

˜ t , φ˜t iF ∗ ,F = 0, 0 ≤ t ≤ T . after checking that η˜t = V˜t − hX

The remaining of this paper is concerned with bond type options, which include caplets on the LIBOR and forward rates. Bond type options We consider a bond type option on PT (µ) with (non-discounted) payoff ξ = g(PT (µ)), maturity S, and discount factor BS−1 .

11

Proposition 3.3 Letting " # " # ˜S (S) ˜ P P (S) S φ˜t (dx) = IE g(PT (µ)) Ft δS (dx) − IE PT (µ)g 0 (PT (µ)) Ft δT (dx) ˜ ˜ Pt (S) Pt (T ) PT (x) + IE P˜S (S)g 0 (PT (µ)) Ft µ(dx), P˜t (x) and η˜t = V˜t − hφ˜t , P˜t iF ∗ ,F , 0 ≤ t ≤ T , yields a self-financing hedging portfolio hedging the claim with payoff ξ = g(PT (µ)). Proof. We have ∞

Z

0

PT (x)(ζt (x) − ζt (T ))µ(dx).

Dt g(PT (µ)) = g (PT (µ)) T

and αt = IE[Dt ξ˜ | Ft ] = IE[Dt (P˜S (S)g(PT (µ)))|Ft ] Z ∞ 0 ˜ ˜ PT (x)(ζt (x) − ζt (T ))µ(dx) Ft = IE[g(PT (µ))Dt PS (S)|Ft ] + IE PS (S)g (PT (µ)) T Z ∞ i h 0 ˜ ˜ (ζt (x) − ζt (T )) IE g (PT (µ))PS (S)PT (x) Ft µ(dx), = ζt (S) IE[PS (S)g(PT (µ))|Ft ] + T

and therefore hαt , dWt iH = IE[P˜S (S)g(PT (µ))|Ft ]ζt (S)dWt Z ∞ IE[g 0 (PT (µ))P˜S (S)PT (x) | Ft ]µ(dx)(ζt (x) − ζt (T ))dWt + "T # P˜S (S) g(PT (µ)) Ft dP˜t (S) = IE P˜t (S) Z ∞ PT (x) 0 ˜ IE g (PT (µ))PS (S) + Ft dP˜t (x)µ(dx) ˜ P (x) T t P (µ) T − IE g 0 (PT (µ))P˜S (S) Ft dP˜t (T ). ˜ Pt (T ) From (2.7) this implies that the process (φ˜t )t∈[0,T ] in (2.1) is given by " # " # ˜S (S) ˜S (S) P P 0 φ˜t (dx) = IE g(PT (µ)) Ft δS (dx) − IE PT (µ)g (PT (µ)) Ft δT (dx) P˜t (S) P˜t (T ) 12

PT (x) 0 ˜ + IE PS (S)g (PT (µ)) Ft µ(dx), P˜t (x) which gives a self-financing hedging portfolio consisting of bonds with maturities S and T , after applying Proposition 2.2 with (Xt )t∈IR+ = (Pt )t∈IR+ .

Bond call options We consider a bond call option on PT (S), S > T , with payoff ξ = (PT (S) − κ)+ and maturity T , and priced at time t ∈ [0, T ] as i h RT − 0 rs ds + ˜ Bt IE[ξ|Ft ] = Bt IE e (PT (S) − κ) Ft h i = Bt IE P˜T (T )(PT (S) − κ)+ | Ft 1 ϑt,T 1 ϑt,T Pt (S) Pt (S) = Pt (S)Φ + − log − κPt (T )Φ log , ϑt,T κPt (T ) 2 ϑt,T κPt (T ) 2 where ϑ2t,T

Z =

T

(ζu (S) − ζu (T ))2 du,

0 ≤ t ≤ T.

t

We have the following corollary of Proposition 3.3. Corollary 3.4 Letting ϑ 1 P (S) 1 P (S) ϑ t,T t t t,T φ˜t (dx) = Φ + log + log δS (dx) − κΦ − δT (dx) 2 ϑt,T κPt (T ) 2 ϑt,T κPt (T ) and η˜t = 0, 0 ≤ t ≤ T , yields a self-financing hedging portfolio for the bond option on PT (S), consisting of bonds with maturities S and T . Proof.

We apply Proposition 3.3 with g(x) = (x − κ)+ , µ(dx) = δS (dx), and the

discount factor BT−1 . We find " # ˜T (T ) P PT (S) ˜ ˜ φt (dx) = −κ IE 1{PT (S)>κ} Ft δT (dx) + IE PT (T )1{PT (S)>κ} Ft δS (dx) P˜t (T ) P˜t (S) 1 Pt (S) ϑt,T 1 Pt (S) ϑt,T = −κΦ log − δT (dx) + Φ log + δS (dx). ϑt,T κPt (T ) 2 ϑt,T κPt (T ) 2 13

Caplets on the LIBOR rate Next, we consider a caplet with payoff ξ = (S − T )(L(T, T, S) − κ)+ = (PT (S)−1 − (1 + κ(S − T )))+ ,

S > T,

(3.5)

and maturity S on the LIBOR rate L(t, T, S) =

Pt (T ) − Pt (S) (S − T )Pt (S)

Its price at time t ∈ [0, T ] is given by i h RS + − 0 rs ds ˜ (L(T, T, S) − κ) Ft Bt IE[ξ | Ft ] = (S − T )Bt IE e i h −1 + ˜ = Bt IE PS (S)(PT (S) − (1 + κ(S − T ))) Ft Pt (T ) 1 ϑt,T log = Pt (T )Φ + ϑt,T (1 + κ(S − T ))Pt (S) 2 Pt (T ) ϑt,T 1 log − . −(1 + κ(S − T ))Pt (T )Φ ϑt,T (1 + κ(S − T ))Pt (S) 2 We have the following corollary of Proposition 3.3. Corollary 3.5 Letting Pt (T ) ϑt,T 1 ˜ + log δT (dx) φt (dx) = Φ ϑt,T (1 + κ(S − T ))Pt (S) 2 1 Pt (T ) ϑt,T −(1 + κ(S − T ))Φ log − δS (dx), ϑt,T (1 + κ(S − T ))Pt (S) 2 and η˜t = 0, 0 ≤ t ≤ T , yields a self-financing hedging portfolio consisting of bonds with maturities S and T , that hedges the claim with payoff (3.5). Proof. Applying Proposition 3.3 with g(x) = (1/x−(1+κ(S −T )))+ , µ(dx) = δS (dx), and the discount factor BS−1 , we get " −1 φ˜t (dx) = IE (PT (S))−1 1

# P˜S (S) Ft δT (dx) {PT (S) >1+κ(S−T )} ˜ Pt (T ) " # P˜S (S) −(1 + κ(S − T )) IE 1{PT (S)−1 >[1+κ(S−T )]} Ft δS (dx) P˜t (S) 14

1 Pt (T ) ϑt,T = Φ log + δT (dx) ϑt,T (1 + κ(S − T ))Pt (S) 2 1 Pt (T ) ϑt,T −(1 + κ(S − T ))Φ log − δS (dx). ϑt,T (1 + κ(S − T ))Pt (S) 2 Caplets on the forward rate Finally, we consider a caplet with payoff ξ = (S − T )(f (T, T, S) − κ)+ ,

S > T,

(3.6)

and maturity S on the forward rate f (t, T, S) = −

log Pt (S) − log Pt (T ) S−T

Its price at time t ∈ [0, T ] is given by i h RS Bt IE[ξ˜ | Ft ] = Bt IE e− 0 rs ds (S − T )(f (T, T, S) − κ)+ Ft i h + ˜ = Bt IE PS (S)(− log PT (S) − κ(S − T )) Ft 2 2 ! ϑt,T 1 Pt (S) ϑt,T + κ(S − T ) + log = Pt (S) √ exp − 2 2ϑt,T 2 Pt (T ) 2π ϑ2t,T Pt (S) 1 Pt (S) ϑt,T −Pt (S) κ(S − T ) + + log Φ − κ(S − T ) + log − . 2 Pt (T ) ϑt,T Pt (T ) 2 We have the following corollary of Proposition 3.3. Corollary 3.6 Letting 2 2 ! ϑt,T ϑ 1 P (S) t,T t φ˜t (dx) = √ exp − 2 + κ(S − T ) + log δS (dx) 2ϑt,T 2 Pt (T ) 2π ϑ2t,T Pt (S) 1 Pt (S) ϑt,T − κ(S − T ) + + 1 + log Φ − κ(S − T ) + log − δS (dx) 2 Pt (T ) ϑt,T Pt (T ) 2 P˜t (S) 1 Pt (S) ϑt,T + Φ − κ(S − T ) + log − δT (dx), ϑt,T Pt (T ) 2 P˜t (T ) and η˜t = 0, 0 ≤ t ≤ T , yields a self-financing hedging portfolio consisting of bonds with maturities S and T , that hedges the claim with payoff (3.6). 15

Proof. Applying Proposition 3.3 with g(x) = (−κ(S − T ) − log x)+ , µ(dx) = δS (dx), and the discount factor BS−1 , we have " # ˜S (S) P φ˜t (dx) = IE (− log PT (S) − κ(S − T ))+ Ft δS (dx) P˜t (S) " # P˜S (S) 1{− log PT (S)>κ(S−T )} Ft δS (dx) − IE P˜t (S) " # P˜S (S) 1{− log PT (S)>κ(S−T )} Ft δT (dx) + IE P˜t (T ) 2 ! Pt (S) ϑ2t,T 1 ϑt,T log + + κ(S − T ) δS (dx) = √ exp − 2 2ϑt,T Pt (T ) 2 2π ϑ2t,T Pt (S) 1 Pt (S) ϑt,T − log + κ(S − T ) + +1 Φ − log + κ(S − T ) − δS (dx) Pt (T ) 2 ϑt,T Pt (T ) 2 P˜t (S) 1 ϑt,T Pt (S) + Φ − + κ(S − T ) − log δT (dx). ϑt,T Pt (T ) 2 P˜t (T )

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